Analysis and finite element approximation of an optimal shape control problem for the steady-state Navier-Stokes equations

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Abstract

An optimal shape control problem for the steady-state Navier-Stokes equations
is considered from an analytical point of view. We examine a rather specific model
problem dealing with 2-dimensional channel flow of incompressible viscous fluid: we
wish to determine the shape of a bump on a part of the boundary in order to minimize
the energy dissipation.

To formulate the problem in a comprehensive manner, we study some properties
of the Navier-Stokes equations. The penalty method is applied to relax the difficulty
of dealing with incompressibility in conjunction with domain perturbations and
regularity requirements for the solutions. The existence of optimal solutions for the
penalized problem is presented.

The computation of the shape gradient and its treatment plays central role
in the shape sensitivity analysis. To describe the domain perturbation and to derive
the shape gradient, we study the material derivative method and related shape
calculus. The shape sensitivity analysis using the material derivative method and Lagrange
multiplier technique is presented. The use of Lagrange multiplier techniques,from which an optimality system is derived, is justified by applying a method from
functional analysis.

Finite element discretizations for the domain and discretized description of the
problem are given. We study finite element approximations for the weak penalized
optimality system. To deal with inhomogeneous essential boundary condition, the
framework of a Lagrange multiplier technique is applied. The split formulation decoupling
the traction force from the velocity is proposed in conjunction with the
penalized optimality system and optimal error estimates are derived.